How to add resolution to a mesh created from a curve
The easiest way to go about this would be to make use of an addon called SIRE outline. Also see this answer.

Once you have installed it. It should appear in the Toolshelf, T. What you should be after early is supporting geometry with good edgeflow.



Use the Inset feature of the addon and offset your outline to the desired amount.



After that, it's just a matter of filling in the even faces and for the rest, you have the supporting geometry near the curved edges so a ngon should suffice.



After that, it is as simple as extruding the model, giving it some depth and chamfering the edges. If you should also need to smooth the model, to keep it sharp simply add edgeloops to the edges or make use of the crease tool.